The hemolytic plaque assay: theory for finite layers

Abstract

We extend the mathematical theory of hemolytic plaque growth to include plaques produced by cells secreting antibodies in layers of finite thickness. Previous theories have assumed that the layer was either two-dimensional or of infinite thickness. By using the method of images we derive an equation for the plaque radius as a function of time for layers of any thickness. We show that at short times and at long times the equation reduces to the appropriate infinite three-dimensional and two-dimensional limiting forms, and obtain expressions for estimating the range of times for which these limiting results are valid. For the liquid monolayer technique we obtain a new limiting result. The equation for the plaque radius is a transcendental equation which we solve numerically for a number of cases of interest. These results illustrate a variety of different features of plaque growth associated with the finite thickness of the layer. Experimental studies are usually carried out in layers whose thicknesses are not standardized. In the assays commonly used the thickness h can vary more than six hundred fold, i.e. 1 × 10 −3 cm ⪅ h⪅ 6.5 × 10 −1 cm. Such variation in h will cause widely different kinetics of plaque growth. For typical plaque experiments of one hour duration the two-dimensional limit is valid when h ⪅ 3 × 10 −3 cm while the infinite thickness limit is valid when h⪆ 10 −1 cm. For thicknesses in between these values the finite layer results must be used.